An efficient allocation of resources is: That combination of inputs,
outputs and distribution of inputs, outputs such that any change in the economy
can make someone better off (as measured by indifference curve map) only by
making someone worse off (pareto efficiency).

Also, efficiency is obtained when there is:

Efficiency in production: Producing the greatest value of goods and services
with given resources.

Efficiency in consumption: Production, consumption, and distribution conforming
to people's values based on willingness-to-pay.

Efficiency of system: All mutual advantageous gains from trade are exhausted.

Also, efficiency is obtained when the three marginal conditions
have been met:

Marginal Condition for Exchange: To attain a pareto maximum, the
marginal rate of substitution between any pair of consumer goods must be the
same for all individuals who consume both goods.

Marginal Condition for Factor Substitution: To attain a pareto maximum,
the marginal rate of technical substitution between any pair of inputs must
be the same for all producers who use both inputs.

Marginal Condition for Product Substitution: To attain a pareto maximum,
the marginal rate of transformation in production must equal the margainal
rate of substitution in consumption for every pair of commodities and for
every individual who consumes both.

Also, efficiency is obtained when property rights are non-attenuated.
That is, they are:

Completely specified

Exclusive

Enforceable and enforced

Transferable

Also, a perfectly competitive market leads automatically to an
efficient distribution of consumer goods among consumers.

Perfect competition leads automatically to an efficient allocation
of production resources among producers.